Transmission Performance Analysis of RV Reducers Influenced by Profile

applied sciences

Article Transmission Performance Analysis of RV Reducers Influenced by Profile Modification and Load

Hui Wang, Zhao-Yao Shi *, Bo Yu and Hang Xu

Beijing Engineering Research Center of Precision Measurement Technology and Instruments, College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, China; [email protected] (H.W.); [email protected] (B.Y.); [email protected] (H.X.) * Correspondence: [email protected]; Tel.: +86-139-1030-7299

Received: 31 August 2019; Accepted: 25 September 2019; Published: 1 October 2019

Featured Application: A new multi-tooth contact model proposed in this study provides an effective method to determine the optimal profile modification curves to improve transmission performance of RV reducers.

Abstract: RV reducers contain multi-tooth contact characteristics, with high-impact resistance and a small backlash, and are widely used in precision transmissions, such as robot joints. The main parameters affecting the transmission performance include torsional stiffness and transmission errors (TEs). However, a cycloid tooth profile modification has a significant influence on the transmission accuracy and torsional stiffness of an RV reducer. It is important to study the multi-tooth contact characteristics caused by modifying the cycloid profile. The contact force is calculated using a single contact stiffness, inevitably affecting the accuracy of the result. Thus, a new multi-tooth contact model and a TE model of an RV reducer are proposed by dividing the contact area into several differential elements. A comparison of the contact force obtained using the finite element method and the test results of an RV reducer prototype validates the proposed models. On this basis, the influence of load on the different modification methods is studied, including a TE, the mechanical performance, and the transmission efficiency. In addition, the proposed reverse profile is particularly suitable for situations with a large clearance and torque. This study provides a reliable theoretical basis for a multi-tooth contact analysis of a cycloid profile modification.

Keywords: RV reducer; contact dynamics; transmission errors; cycloid mechanism

1. Introduction RV reducers are planocentric reduction mechanisms used for precise transmission applications. Such mechanisms incorporate numerous cycloid-pin teeth that are engaged concurrently. Therefore, RV reducers have a compact and highly rigid construction that is robust against shock-load and overloading. Furthermore, a minimal backlash and inertia assure a rapid acceleration and extremely accurate positioning. An RV reducer is ideally suited for precision mechanical control in industrial robots, machine tools, and assembly equipment, where precise positioning and a torsional stiffness capability are required [1]. Transmission errors (TEs) are key parameters used to represent the transmission performance of an RV reducer. The TE measurement method of a precision planetary cycloid reducer is defined in GB/T 37718-2019 [2]. The optimal measurement speed can be determined according to the Stribeck friction model [3,4]. The determination of optimal measurement speed of transmission errors could improve the measurement precision. A cycloid profile modification is one of the factors causing the TEs of an RV reducer [5–7]. However, to avoid the influence of machining errors on an RV reducer assemble, a cycloid

Appl. Sci. 2019, 9, 4099; doi:10.3390/app9194099 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 4099 2 of 19 tooth profile modification is commonly used in engineering practice. The modification methods often include a modification of the moved distance, equidistance, rotation angle, and their combination [8]. An angular modification needs to be used with other methods concurrently. In [9], based on a tooth contact analysis, the TE caused by different combinations of modifications is quantitatively analyzed. To reduce the impact of a modification on the RV reducer transmission accuracy, the optimal cycloidal tooth profile modification method has been studied by numerous researchers. In [10], a modification method for designing the modification gap curves by adjusting the positions of five key points on the cycloid profile was proposed. This method can improve the carrying capacity of cycloidal drive, eliminate noise and vibration and develop the transmission accuracy. Compared with the designed modified tooth profile, the actual tooth profile with machining errors can be accurately measured to compensate such errors between the actual and designed tooth profiles [11,12]. In addition, judging the optimal modified tooth profile according to the amplitude frequency vibration response of the transmission system caused by different modified tooth profiles is an effective method [13]. The torsional stiffness of an RV reducer achieves a relatively large benefit from its multi-tooth contact characteristics, and thus can ensure a high transmission accuracy. Therefore, the contact force calculation of a cycloid-pin is particularly important when multiple teeth mesh together to transmit a power load [14,15]. Among the many studies conducted in this area, Xu proposed a numerical method to determine the number of teeth that are loaded simultaneously [16,17]. The model can be used to investigate the effect of changing the geometrical design parameters of the tooth profile of the cycloid gear. However, the contact area of each mesh tooth is treated as a whole, and the contact force is calculated using a single contact stiffness. Such a treatment method results in some errors in the actual contact status [18]. To reduce this deficiency, Ma established a hybrid contact force model. The contact cylinder surface is divided into many differential elements using the discrete element theory. Each differential element calculates the contact force separately according to the compression depth. Therefore, the contact model is more accurate for a complex profile contact [19]. Studies on cycloid tooth profile modification have mostly focused on the design method of optimizing the tooth profile, and the evaluation index is often limited to the transmission accuracy. The influences of the modification of the mechanical properties, contact area, and transmission efficiency have rarely been reported. However, such performances are also of concern among users of an RV reducer. In addition, the most important point that is often neglected is the load impact on the tooth profile modification. Regarding the multi-tooth contact model introduced in the references herein, the accuracy of the results will be significantly affected because it is based on a single contact stiffness model. In this paper, each tooth contact area is replaced by multiple differential contact units, and a multi-tooth contact model and a TE model of an RV reducer are first established. The proposed models are then verified using the finite element method and experimental measurements. On this basis, the influence of load on the different modification methods is studied in detail. The evaluation index considered includes the transmission accuracy, mechanical performance, and transmission efficiency. Finally, considering the load influence, the proper applications of different modification methods are discussed.

2. Analytical Model of RV Reducer with Tooth Proﬁle Modiﬁcations

2.1. Contact Force Calculation

2.1.1. Contact Force Model A traditional cycloid drive is the basis used to analyze an RV reducer, and Figure1 shows a traditional cycloid drive coordinate system. Here, xoy and xcoc yc represent the static coordinates of a pin wheel and cycloid gear, O and Oc denote the center of the pin wheel and cycloid gear, and rp and rc denote the pitch circle radius of a pin wheel and cycloid gear, respectively. Pitch point P is the instant velocity center of a cycloid drive and OOc is the position of the crank. When the crank rotates Appl. Sci. 2019, 9, 4099 3 of 19

Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 19 θcr, the cycloid gear will rotate θcy according to the transmission ratio. The rotation of a cycloid gear is θ θ outputrotates using cr , the a parallelogramcycloid gear will mechanism. rotate cy accor

Figure 1. Traditional cycloid drive coordinates. Figure 1. Traditional cycloid drive coordinates. The track of the meshing point of a cycloid-pin in a static coordinate system is part of the circular arc ofThe the track pin. According of the meshing to the meshingpoint of principle a cycloid-pin of a cycloid in a static drive, coordinate the tooth profile system of is a cycloidpart of gear the cancircular be obtained arc of the as pin. follows According [20]: to the meshing principle of a cycloid drive, the tooth profile of a cycloid gear can be obtained as follows [20]:  i 1 n o H H H H  x = Rp + ∆Rp cos[ 1 i )θ a cos(i θ) + rrp + ∆rrp S− 2 Kcos(i θ) cos[ 1 i θ]  𝑥=(𝑅 +𝛥𝑅 )𝑐𝑜𝑠[(1−𝑖 −)𝜃]−𝑎𝑐𝑜𝑠(𝑖i − 𝜃) + (𝑟 +𝛥𝑟 )𝑆 1𝐾𝑐𝑜𝑠(𝑖n 𝜃)− − 𝑐𝑜𝑠[ − (1−𝑖 )𝜃]o , (1)  H H H H  y = Rp + ∆Rp sin[ 1 i )θ + a sin(i θ) rrp + ∆rrp S − 2 Ksin(i θ) + sin[ 1 i θ] , (1) − − − 𝑦=(𝑅 +𝛥𝑅)𝑠𝑖𝑛[(1−𝑖 )𝜃]+𝑎𝑠𝑖𝑛(𝑖 𝜃) − (𝑟 +𝛥𝑟)𝑆 𝐾𝑠𝑖𝑛(𝑖 𝜃) + 𝑠𝑖𝑛[ (1−𝑖 )𝜃] where where Zp iH = , (2) Zc H Z p i = , (2) aZZpc K = , (3) Rp + ∆Rp aZ p S = 1K+=K2 2Kcos θ, .(3) (4) RR−pp+Δ Here, θ represents the angle of the pin position vector; ∆Rp and ∆rrp refer to the modification coefficient of the moved distance and equidistance,𝑆=1+𝐾 respectively;−2𝐾𝑐𝑜𝑠𝜃.Zc and Zp denote the cycloid disc teeth(4) number and pin number; rrp and Rp represent the pin radius and pin center circle radius, respectively; Here,𝜃represents the angle of the pin position vector; ΔRp and Δrrp refer to the modification and a represents the eccentricity. coefficient of the moved distance and equidistance, respectively; 𝑍 and 𝑍 denote the cycloid disc The pitch circle radius of a pin wheel and cycloid gear can be calculated using the following teeth number and pin number; rrp and Rp represent the pin radius and pin center circle radius, equations: respectively; and 𝑎 represents the eccentricity.( rp = aZp The pitch circle radius of a pin wheel and cycloid gear can be calculated using the following(5) rc = aZc equations: Figure2 shows the mechanism of an RV reducer. Because the machining precision of the crank, = cycloid gear, and planetary carrier is extremely high,raZp thesep are generally known as the three key parts  = (5) of an RV reducer. Two cycloid gears are supported raZcc in the planet carrier by three cranks. The purpose of these two cycloid gears is to balance the radial force and improve the load capacity. An RV reducer containsFigure three 2 shows freedoms the ofmechanism rotation, namely, of an RV the reducer. rotations Because of the inputthe machining gear shaft, precision pin wheel, of andthe planetcrank, carrier.cycloid Anygear, fixedand planetary parts can formcarrier a mechanismis extremely for high, increasing these are or generally decreasing known the speed. as the Because three key an parts of an RV reducer. Two cycloid gears are supported in the planet carrier by three cranks. The purpose of these two cycloid gears is to balance the radial force and improve the load capacity. An RV reducer contains three freedoms of rotation, namely, the rotations of the input gear shaft, pin wheel, and planet carrier. Any fixed parts can form a mechanism for increasing or decreasing the Appl. Sci. 2019, 9, 4099 4 of 19 Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 19 speed.involute Because planet an drive involute is applied planet to drive the high-speedis applied to stage, the high-speed its TE is significantly stage, its TE attenuated is significantly by the attenuatedtransmission by chain the transmission [5,6]. Therefore, chain to focus [5,6]. on Theref the mainore, factorsto focus aff ectingon the the main transmission factors affecting performance the transmissionof an RV reducer, performance an involute of an gear RV transmission reducer, an involute is not considered gear transmission in the present is not study. considered The torque in the is presentapplied study. to the The three torque cranks is asapplied an input to the of thethree involute cranks planetaryas an input drive. of the The involute RV reducer planetary parameters drive. Theused RV in reducer this paper parameters are listed used in Table in this1. paper are listed in Table 1.

FigureFigure 2. 2. ConstructionConstruction of of RV RV reducer. reducer.

Table 1. Parameters of RV reducer. Table 1. Parameters of RV reducer.

ParametersParameters ValueValue TransmissionTransmission ratio ratio 81 81 Pin center circle radius R (mm) 114.5 Pin center circle radius Rp （mm)p 114.5 Crank center circle radius Rz (mm) 70 𝑅 Crank center circlePin radius radiusrrp (mm) (mm) 5 70 Pin radiusEccentricityrrp (mm)a (mm) 2.2 5 EccentricityPin number𝑎 (mm)Z p 402.2 Number of cycloid disc teeth Zc 39 Z p WidthPin number of cycloid disc L (mm) 1840 NumberRated of cycloid output disc torque teethTo (NZcm) 78439 · 5 WidthElasticity of cycloid modulus disc of𝐿 pin(mm)Ep (MPa) 2.19 1810 × Poisson’s ratio of pin vp 0.3 Rated output torque𝑇 (N·m) 784 5 Elasticity modulus of cycloid disc Ec (MPa) 2.11 10 Elasticity modulus of pin E p (MPa) 2.19× × 105 Poisson’s ratio of cycloid disc vc 0.292 Poisson’sModification ratio parameter of pin vp∆ rrp (mm) 0.050.3 Modification parameter ∆R (mm) 0.05 Elasticity modulus of cycloid disc Epc (MPa) 2.11 × 105 Poisson’s ratio of cycloid disc vc 0.292 A geometric modelModification of an RV parameter reducer isΔ shownrrp (mm) in Figure 3. Here, O and0.05O1 represent the center of a pin wheel and cycloid gear 1, respectively. In addition, pitch point P represents the instant velocity Modification parameter ΔRp (mm) 0.05 center of a cycloid gear 1, Opi denotes the center of the ith pin, ci refers to the meshing point of the ith pin with cycloid gear 1, and li represents the normal arm length. Moreover, ∆OPO1 is defined as A geometric model of an RV reducer is shown in Figure 3. Here, O and 𝑂 represent the center the meshing triangle, where φ , ψ , and θ denote the contact angle, normal angle, and azimuth of of a pin wheel and cycloid geari 1,i respectively.i In addition, pitch point 𝑃 represents the instant the ith pin. Because three cranks are supported in the planet carrier, as a difference between the RV velocity center of a cycloid gear 1,𝑂denotes the center of the ith pin,𝑐refers to the meshing point of reducer and a traditional cycloid drive, the cranks have an extra rotation relative to the pin wheel. the ith pin with cycloid gear 1, and 𝑙represents the normal arm length.H Moreover, 𝛥𝑂𝑃𝑂 is defined The angle between the center line OO1 and the X axis is ϕ = θcr/i , where θcr is the crank rotation as the meshing triangle, where 𝜙, 𝜓, and 𝜃 denote the contact angle, normal angle, and azimuth angle. The rotation angle of the planet carrier is θ = θ /Z . of the ith pin. Because three cranks are supportedpl in thecr planetp carrier, as a difference between the RV reducer and a traditional cycloid drive, the cranks have an extra rotation relative to the pin θ wheel. The angle between the center line 𝑂𝑂 and the 𝑋 axis is 𝜑=𝜃/𝑖 , where cr is the crank rotation angle. The rotation angle of the planet carrier is θθpl= cr/Z p . Appl. Sci. 2019, 9, 4099 5 of 19 Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 19

FigureFigure 3. 3.Geometric Geometric model model of of RV RV reducer. reducer.

TheThe contact contact angle angle can can be be obtained obtained based based on on the the geometric geometric relationship relationship of of∆ OPO𝛥𝑂𝑃𝑂pi, which, which can can be be expressedexpressed as as Equation Equation (6) (6) [21 [21]:]:

" # " # 𝜙 =𝑡𝑎𝑛1 aZpsin αi =𝑡𝑎𝑛1 sin αi , (6) φi = tan− = tan− /( ) , (6) Rp aZp cos αi RP/(aZp) cos αi − − 𝛼 =𝜑− ⋅𝑖 i iZ= 1,2,3... p where , and is the pin number, i.e., . 2π where αi =ithϕ H i, and i is the pin number, i.e., i = 1, 2, 3 ... Zp. The −pini azimuth· may be obtained using Equation (7). The ith pin azimuth may be obtained using Equation (7). 2𝜋 𝜃 =𝑚𝑜𝑑( ⋅𝑖−𝜑,𝜋) (7) 2π𝑍 θi = mod( i ϕ, π) (7) Zp · − Furthermore, the normal angle 𝜓 can be obtained from the inner angle relationship of 𝛥𝑂𝑃𝑂.

Furthermore, the normal angle ψi can be𝜓 obtained =𝜋−𝜃 from −𝜙 the inner angle relationship of ∆OPOpi(8). The static coordinate value of the theoretical meshing point 𝑐 may be obtained through a ψi = π θi φi (8) coordinate transformation as follows: − −

The static coordinate value of the theoretical meshing point ci may be obtained through a coordinate 𝑥 𝑐𝑜𝑠 𝜃 −𝑠𝑖𝑛𝜃 𝑎𝑐𝑜𝑠(𝜃/𝑖 ) 𝑥 transformation as follows: 𝑦 𝑦 =𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 𝑎𝑠𝑖𝑛(𝜃/𝑖 ) , (9) 1 1    00( 1H)    xpi   cos θpl sin θpl a cos θcr/i  xi     − H   Where 𝑥 and 𝑦 are the yvaluespi  = in sinEquationθpl cos (1),θ plwhicah sin may(θcr be/i calculated)  yi , by substituting 𝛼 into (9) 𝜃.      The contact deformation 1  caused by0 torque 0 load To between 1 the cycloid1  gear 1 and the ith pin is

𝛿 in the enlarged view of Figure 3. As a result of a contact deformation, the cycloid gear 1 will rotate where xi and yiΔareβ the values in Equation (1), which may be calculated by substituting αi into θ. a small angle around 𝑂 in the direction of the torque load. The contact deformation caused by torque load To between the cycloid gear 1 and the ith pin is δi The contact deformation𝛿may be expressed as follows: in the enlarged view of Figure3. As a result of a contact deformation, the cycloid gear 1 will rotate a small angle ∆β around O1 in the direction of the𝛿 torque=𝑙𝛥𝛽 load. −𝐸 (10)

Here, 𝐸 is the initial tooth gap along the normal line caused by a tooth profile modification (TPM), and is calculated as follows [20]: Appl. Sci. 2019, 9, 4099 6 of 19

The contact deformation δi may be expressed as follows:

δ = l ∆β E (10) i i − i

Here, Ei is the initial tooth gap along the normal line caused by a tooth proﬁle modiﬁcation (TPM), and is calculated as follows [20]: p   2  sin θ  1 K cos θi 1 K sin θi E ( ) = r  i  R i θi ∆ rp1 p  ∆ p − p − − (11) − 1 + K2 2K cos θ − · 1 + K2 2K2cos θ − i − i

The normal arm length li at meshing point ci may be calculated using Equation (12).

li = rc sin ψi (12)

If δ 0, the pin will mesh with cycloid gear 1; otherwise, the pin will not participate in the i ≥ transmission. This condition is used to judge the number of pins that are simultaneously engaged. To calculate the contact force more accurately, the Gaussian quadrature method is used to discretize the contact areas in the present study [22]. The compression region is divided into many small differential units, as indicated in the enlarged view of Figure3. Thus, each di fferential element can be replaced with a spring-damped model. Therefore, the normal contact force of the contact area can be expressed as follows: Xn F H F(δ ), (13) Ni ≈ j ij j=1 where n represents the number of differential elements, and Hj is the weight coefficient. As shown in Figure3, the penetration of the jth small element can be calculated based on the difference between rrp and OpiB as follows: δ = rrp O B (14) ij − pi The area of the differential element may be calculated in the following manner:

1 dA = r Ldτ (15) ij 2 rp Because the Lankarani–Nikravesh contact force model exhibits a higher accuracy based on the Hertz contact force model, it is adopted to calculate the contact force of a single differential element. The contact force model is as follows [23]: . n dFNij = dKijδij + dDijδij, (16) where Kij is the Hertz contact stiffness of the differential element, which is calculated as follows [19]:

EcEpAij Kij = (17) h 2 2i Ec 1 v + Ep 1 v W − c − p ij

Here, Wij is the thickness of the elastic layer, and Dij is the damping coefficient, the modified formula of which is [24]: ( 2) 2(1 ce) n 3Kij 1 ce e − δij D = − ( 0) (18) ij . ( ) δij > − 4δij . ( ) − where δij is the initial penetration velocity upon impact of the differential element of the ith contact area, and ce is the restitution coefficient. The exponent n is set to 1.5 for metallically circular or elliptical surfaces. Appl. Sci. 2019, 9, 4099 7 of 19

Thus, the contact force of the entire contact area of the ith pin is as follows:

τi τi . ( ) . Z Z − 2 2(1 ce) 1.5 4δij + 3(1 ce )e − δij rrpδij LEcEp FNi = dFNij = ( − ) dτ (19) . ( ) h 2 2i − Ec 1 v + Ep 1 v 0 0 8Wijδij − c − p

The contact force FNi can be divided into the radial and tangential components in the direction and vertical direction of the center line OO1. The contact force can then be expressed as follows:

t r FNi = FNii + FNi j (20) ( Ft = F sinψ Ni Ni i (21) r = FNi FNicosψi In this study, the Ambrósio friction force model is adopted to calculate the friction force between meshing surfaces. Compared with the traditional Coulomb friction model, the dynamic correction coefficient cd is added to the Ambrósio friction model [25], effectively preventing the friction from changing direction when the tangential velocity is zero. Therefore, the improved friction model creates numerical stability and is more suitable for a computer simulation. The calculation formula is as follows: V f y F = c c F = Fx i + F j, (22) f i − f d Ni V f i f i k f k   Fx = F sin(ϕ ψ )  f i f i i  y − − , (23)  F = F f i cos(ϕ ψi) f i − x y where F f i and F f i represent the horizontal and vertical components of friction, respectively; c f is the friction coefficient; V f is the relative tangential velocity; and cd is the dynamic correction coefficient, which can be expressed in the following manner:  0 i f V f vd  k k ≤  V f vd cd =  k k− i f vd V f vu , (24)  vu vd ≤ k k ≤  − 1 i f V vu k f k ≥ where vd and vu are the given tolerances for the relative tangential velocity of the contact surfaces. Then, the torque balance equation of cycloid transmission may be expressed as follows:

m .. X2 t x y To Jθ = 2 F (r0 + a) F y0 + F x0 , (25) − pl Ni c − f i· pi f i· pi i=m1 where m1 is the serial number of pins that begin to mesh, and m2 is the serial number of pins that exit the meshing, which are determined from Equation (10). The simulation ﬂow of all calculation processes is shown in Figure4. Appl. Sci. 2019, 9, 4099x FOR PEER REVIEW 88 of of 19

Figure 4. Simulation flow chart. Figure 4. Simulation flow chart. 2.1.2. Contact Force Model Verification Using the Finite Element Method 2.1.2. Contact Force Model Verification Using the Finite Element Method The dynamics simulation model of an RV reducer is established according to the parameters listedThe in Tabledynamics1. This simula dynamicstion model analysis of an employs RV reducer SolidWorks is established 2017 COSMOS according (Dassault to the parameters Systemes, Paris,listed in France) Table for1. This modeling dynamics and solving.analysis employs The stress SolidWorks variationof 2017 an RVCOSMOS reducer (Dassault during movement Systemes, canParis, be France) calculated for modeling using the and finite solving. element The method stress variation (FEM). Figure of an 5RVa shows reducer the during FE model movement of an RVcan reducer,be calculated and Figure using5 bthe shows finite the element mesh divisionmethod model(FEM). of Figure a cycloid 5a gearshows and the pin FE wheel. model To of simplify an RV thereducer, simulation and Figure model, 5b theshows pins the are mesh fixed division on the wheel.model of In a addition,cycloid gear the and non-pin pin wheel. wheel To can simplify reduce vibrationsthe simulation and stressmodel, fluctuations the pins are [26 ].fixed For on consistency the wheel. with In theaddition, theoretical the non-pin model, the wheel influence can reduce of the firstvibrations stage isand not stress considered. fluctuations The rotation [26]. For motion consistency is applied with to the the theoretical three cranks model, individually, the influence and the of loadthe first torque stage is is applied not considered. to the center The ofrotation the planet motion carrier. is applied The influence to the three of thecranks bearing individually, is simulated and bythe settingload torque the contact is applied stiff toness the and center damping of the plan coeffietcient carrier. at theThe bearing influence bore. of the The bearing impact is function simulated is usedby setting to calculate the contact the contact stiffness force an iteratively.d damping According coefficient to at the the lubrication bearing bore. condition The impact of the RVfunction reducer is prototype,used to calculate the static the friction contact coe forcefficient iteratively. between According the relative to moving the lubrication surfaces iscondition set as 0.08, of andthe theRV dynamicreducer prototype, friction coe thefficient static is friction set as 0.05. coefficient between the relative moving surfaces is set as 0.08, and theIn thisdynamic study, friction the total coefficient simulation is set time as 0.05. is 20 s, and the data are obtained from a period of 0.0375–18.075 s. The results indicate that the crank rotates one round, the data are solved at every angle, the interval angle is 1/10◦, and the interval time is 0.00625 s. The contact stress on the cycloid tooth surface as the crank rotates 170◦, is shown in Figure6. Appl.Appl. Sci.Sci. 20192019,, 99,, xx FORFOR PEERPEER REVIEWREVIEW 99 ofof 1919

Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 19 Appl. Sci. 2019, 9, 4099 9 of 19

(b) ((aa)) (b)

FigureFigure 5.5. FiniteFinite elementelement (FE)(FE) modelmodel ofof RVRV reducer:reducer: ((aa)) explodedexploded viewview andand ((bb)) meshmesh calculation.calculation.

InIn thisthis study,study, thethe totaltotal simulationsimulation timetime isis 2020 s,s, andand thethe datadata areare obtainedobtained fromfrom aa periodperiod ofof 0.0375–0.0375– 18.07518.075 s.s. TheThe resultsresults indicateindicate(a) thatthat thethe crankcrank rotatesrotates oneone round,round, thethe datadata areare ( solvedbsolved) atat everyevery angle,angle, thethe intervalinterval angleangle isis 1/10°,1/10°, andand thethe intervalinterval timetime isis 0.006250.00625 s.s. TheThe contactcontact strestressss onon thethe cycloidcycloid toothtooth FigureFigure 5.5. Finite elementelement (FE)(FE) modelmodel ofof RVRV reducer:reducer: ((aa)) explodedexploded viewview andand ((bb)) meshmesh calculation.calculation. surfacesurface asas thethe crankcrank rotatesrotates 170°,170°, isis shownshown inin FigureFigure 6.6. In this study, the total simulation time is 20 s, and the data are obtained from a period of 0.0375– 18.075 s. The results indicate that the crank rotates one round, the data are solved at every angle, the interval angle is 1/10°, and the interval time is 0.00625 s. The contact stress on the cycloid tooth surface as the crank rotates 170°, is shown in Figure 6.

Figure 6. Contact stress on the cycloid tooth surface during meshing. FigureFigure 6.6. ContactContact stressstress onon thethe cycloidcycloid toothtooth surfacesurface duringduring meshing.meshing.

To verify the validity of the analytical model for a profile-shifted gear with TPM, ∆rrp = 0 to 0.08 To verify the validity of the analytical model for a profile-shifted gear with TPM, 𝛥𝑟𝛥𝑟 =0=0 to mm isTo adopted verify the to calculatevalidity of the the contact analytical force model and friction for a profile-shifted which, as obtained gear throughwith TPM, analytical and to 0.080.08 mmmm isis adoptedadopted toto calculatecalculate thethe contactcontact forceforce andand frictionfriction which,which, asas obtainedobtained throughthrough analytical analytical FE methods, are displayed in Figure7. The contact force and friction at a rotation angle of 135 ◦ are andand FEFE methods,methods, areare displayeddisplayed inin FigureFigure 7.7. TheThe contactcontact forceforce andand frictionfriction atat aa rotationrotation angleangle ofof 135°135° compared, demonstratingFigure 6. the Contact largest stress differences. on the cycloid The percentage tooth surface of during differences meshing. between the analytical areare compared,compared, demonstratingdemonstrating thethe largestlargest differencedifferences.s. TheThe percentagepercentage ofof differencesdifferences betweenbetween thethe methodanalytical and method the FE and method the areFE method presented are in presented Table2. The in resultsTable 2. from The theresults analytical from the and analytical FE methods and analyticalTo verify method the validityand the FEof themethod analytical are presented model for in a Table profile-shifted 2. The results gear fromwith theTPM, analytical 𝛥𝑟 =0 and to areFE inmethods good agreement. are in good agreement. 0.08FE methods mm is adopted are in good to calculate agreement. the contact force and friction which, as obtained through analytical and FE methods, are displayed in Figure 7. The contact force and friction at a rotation angle of 135° are compared, demonstrating the largest differences. The percentage of differences between the analytical method and the FE method are presented in Table 2. The results from the analytical and FE methods are in good agreement.

((aa)) ((bb))

Figure 7. Cont.

(a) (b) Appl. Sci. 2019, 9, 4099 10 of 19 Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 19

FigureFigure 7.7.Contact Contact force force and and friction friction obtained obtained using using analytical analytical (right) (right) and FE and (left) FE methods: (left) methods: (a,b) contact (a, b) forcecontact and force (c,d )and friction. (c, d) friction.

Table 2.2. PercentagePercentage ofof didifferencesﬀerences betweenbetween analyticalanalytical andand FEFE method.method.

ContactContact Force force (N) (N) Relative FrictionFriction (N)(N) Relative ModificationModification Relative Relative AnalyticalAnalytical Didifferencefference AnalyticalAnalytical Didifferencefference Parameterparameter FEFE Methodmethod FE Methodmethod Methodmethod (%)(%) Methodmethod (%) 00 550 550 575575 4.544.54 5353 51 3.77 0.020.02 582582 603603 3.613.61 5656 55 1.78 0.040.04 625625 655655 4.814.81 5858 59 1.72 0.06 673 685 1.78 62 61 1.61 0.06 673 685 1.78 62 61 1.61 0.08 701 725 3.42 65 66 1.53 0.08 701 725 3.42 65 66 1.53 2.2. Tranmission Error Calculation 2.2. Tranmission Error Calculation 2.2.1. Transmission Error Model of RV Reducer 2.2.1. Transmission Error Model of RV Reducer A TE is one of the most important methods used to measure the accuracy of an RV reducer, and is definedA TE as theis one diff oference the most between important the theoretical methods used and actual to measure rotation the angles accuracy of the of an output RV reducer, shaft as and the inputis defined shaft as rotates the difference at any angle. between Because the thistheoretical study focuses and actual on the rotation influence angles of the of cycloidthe output drive shaft in anas RVthe reducer,input shaft the rotates angle of at the any crank angle. is expressedBecause this as thestudy input focuses angle. on Thus, the influence the TE may of the be calculatedcycloid drive as follows:in an RV reducer, the angle of the crank is expressed as the input angle. Thus, the TE may be calculated as follows: i i i θ = θ /Zp θ (26) e cr − pl 𝜃 =𝜃 /𝑍 −𝜃 Here, the upper corner i indicates a certain moment, and the mean value of the TE is the following:(26) Here, the upper corner 𝑖 indicates a certain moment, and the mean value of the TE is the Xn following: 1 i θe = θe, (27) n 𝜃 = i=∑1 𝜃, (27) where n represents the number of data points. In addition, the TE amplitude is calculated as follows: where 𝑛 represents the number of data points. In addition, the TE amplitude is calculated as follows: a i i θe = max(θe) min(θe) (28) − 𝜃 =𝑚𝑎𝑥(𝜃)−𝑚𝑖𝑛(𝜃) (28) Because the main factor generating the mean value of the TE is the backlash of the RV reducer, the meanBecause value of the the main TE can factor be used generating to calculate the mean indirectly value the of the backlash TE is ofthe the backlash RV reducer. of the As RV the reducer, mean valuethe mean of the value TE also of the includes TE can the becontact used to deformation calculate indirectly caused bythe a backlash tooth engagement, of the RV thisreducer. part canAs the be approximatedmean value of based the TE on also the includes amplitude the of contact the TE. deformation Additionally, caused considering by a tooth that the engagement, starting position this part of thecan cycloid be approximated gear is in the based middle on ofthe the amplitude tooth gap, of the the backlash TE. Additionally, of the RV reducer considering may bethat approximated the starting asposition follows: of the cycloid gear is in the middle of the tooth gap, the backlash of the RV reducer may be approximated as follows: a B 2θe θe (29) ≈ − 𝐵2𝜃 −𝜃 (29)

2.2.2. Transmission Error Measurement Experiment Appl. Sci. 2019, 9, 4099 11 of 19

2.2.2.Appl. Transmission Sci. 2019, 9, x FOR Error PEER MeasurementREVIEW Experiment 11 of 19 Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 19 AA physical physical prototype prototype of anof an RV RV reducer reducer is produced is produced according according to the to parametersthe parameters in Table in Table1. Several 1. testsSeveral wereA testsphysical conducted were prototype conducted to measure of to an themeasure RV transmission reducer the transmission is performanceproduced performance according of the RV to of reducerthe the parametersRV prototypereducer inprototype toTable validate 1. thetoSeveral analytical validate tests the model. were analytical conducted These model. tests to Thesemeasure were tests conducted the were transmission conducted on a rig performance on composed a rig composed of the a servo RVof a reducer servo motor, motor, prototype two torquetwo meterstorqueto validate to meters measure the to analytical measure the input model. the and inputoutput These and tests torque,output were to two rque,conducted round two round gratings on a rig gratings composed to measure to measure of the a servo input the motor, input and output andtwo rotationoutputtorque angles, rotationmeters the to angles, measure tested the reducer, the tested input and reducer, and a magneticoutput and toarque, brakemagnetic two acting roundbrake as agratingsacting load equipment.as to a measureload equipment. The the ratedinput torqueTheand ofrated theoutput magnetic torque rotation of brake the angles, magnetic is 2000 the tested Nbrakem. Theisreducer, 2000 test N rig·andm. isThe a shown magnetic test rig in is Figure brake shown 8acting. in The Figure controlas a 8.load The system equipment. control is shownsystem The in · Figureisrated shown9. torque The in input Figureof the and magnetic 9. output The input brake round and is gratings 2000 output N·m. are ro Theund mounted test gratings rig onis shown theare endmounted in of Figure the on RV 8. theThe prototype. end control of the Tosystem avoidRV theprototype.is inﬂuence shown inTo of Figure eccentricityavoid the9. Theinfluence on input the accuracyof and eccentricity output of the ro on outputund the gratings accuracy grating, are of the themounted thin outp reedut on grating, structure the end the is ofthin adopted the reed RV to connectstructureprototype. the is output Toadopted avoid shaft tothe connect and influence the the round of output eccentricity grating. shaft The andon the measurementthe accuracy round grating. of accuracythe outpThe utmeasurement of grating, the test the rig accuracythin is 2 reed arcsec. structure is adopted to connect the output shaft and the round grating. The measurement accuracy Theof outputthe test shaftrig is of2 arcsec. the RV The reducer output was shaft loaded of the to RV 784 reducer N m using was loaded a magnetic to 784 brake. N·m using The parametersa magnetic of of the test rig is 2 arcsec. The output shaft of the RV reducer· was loaded to 784 N·m using a magnetic thebrake. sensors The are parameters listed in Tableof the3 sensors. are listed in Table 3. brake. The parameters of the sensors are listed in Table 3.

FigureFigure 8.8. TransmissionTransmission performance tester tester of of RV RV reducer. reducer. Figure 8. Transmission performance tester of RV reducer.

Figure 9. Control system of transmission performance tester. Figure 9. Control system of transmission performance tester. Figure 9. Control system of transmission performance tester. Table 3. Parameter list of sensors. Table 3. Parameter list of sensors. Appl. Sci. 2019, 9, 4099 12 of 19

Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 19 Table 3. Parameter list of sensors.

NameName Type Type Range Range AccuracyAccuracy TorqueTorque sensors sensors HY-G33/ HY-G33 HY-G32/HY-G32 ±10 N10·m/±1500 N m/ 1500 N·m N m ±0.25%0.25% FsFs ± · ± · ± RoundRound gratings gratings REXM REXM 360° 360 2±2"/±1"/ 1 ◦ ± 00 ± 00

TheThe forward andand backward backward TEs TEs were were tested tested in onein one loop loop to obtain to obtain the backlash the backlash of the RVof reducer.the RV reducer.Moreover, Moreover, the TE in the one TE direction in one direction was tested was twice test continuouslyed twice continuously to verify theto verify repeatability the repeatability of the test ofresults. the test It canresults. be seen It can from be Figureseen from 10a thatFigure the 10a TE isthat consistent. the TE is Theconsistent. backlash The of thebacklash RV prototype of the RV is prototype31.03 arcsec. is In31.03 addition, arcsec. the In analytical addition, backlash the analytic is 28.12al arcsec,backlash as calculatedis 28.12 arcsec, using Equationas calculated (29), whichusing Equationis smaller (29), than which the experimental is smaller than value the because experime smallntal inﬂuencing value because factors small are influencing ignored. The factors forward are ignored.transmission The forward error is 40transmission arcsec, and error the backwardis 40 arcsec TE, and is 42 the arcsec. backward Figure TE 10 isb 42 shows arcsec. the Figure harmonic 10b showsspectrum the ofharmonic the forward spectrum TE as theof the output forward shaft TE rotates as the one output round. shaft Figure rotates 10c,d one show round. an enlarged Figure 10c,d view showof the an forward enlarged and view backward of the TEs, forward respectively, and backward as the cranks TEs, rotate respectively, six rounds. as the cranks rotate six rounds.

(a) (b)

FigureFigure 10. 10. TestTest results of RV prototype: ( a) transmission error (TE) including backlash, ( b) harmonic spectrumspectrum of of a a TE as output shaft rotates one round, ( c) enlarged view of forward TE in the dashed dashed frameframe of of ( (aa),), and and ( (d)) enlarged enlarged view of backward TE in the dashed frame of (a).

AsAs the cranks rotaterotate oneone round,round, the the TE TE curves curves generate generate a largea large peak. peak. There There are are 40 40 main main peaks peaks as theas theoutput output shaft shaft rotates rotates one round, one round, which iswhich the same is the as thesame crank as rotationthe crank period. rotation This period. fully demonstrates This fully demonstratesthat the cycloid that drive the cycloid is the main drive factor is the amainffecting factor the affecting transmission the transmission performance performance of the RV reducer, of the RVwhich reducer, also indicates which thealso correctness indicates ofthe the correctness simplified model.of the Mostsimplified of the othermodel. harmonics Most of are the multiple other harmonicsfrequencies are of themultiple cranks, frequencies as shown inof Figurethe cranks, 10b. Theas shown details in of Figure the main 10b. harmonic The details components of the main are harmoniclisted in Table components4. Because are the listed cycloid in Table gear generates4. Because 39 the meshing cycloid cyclesgear generates in one crank 39 meshing revolution, cycles it can in onebe concluded crank revolution, that the 1560thit can be harmonic concluded is caused that the by the1,560th cycloid harmonic tooth profile is caused [27]. by The the TE cycloid amplitude tooth is profile0.47 arcsec, [27]. whereasThe TE theamplitude analytical is result0.47 arcsec, is 0.49 arcsec.whereas The the percentages analytical ofresult diff erencesis 0.49 betweenarcsec. The the percentagesanalytical method of differences and the between test results the areanalytical presented method in Table and5 .the The test analytical results are and presented test results in areTable in 5.good The agreement.analytical and test results are in good agreement.

Table 4. Main harmonic components of the test TE. Appl. Sci. 2019, 9, 4099 13 of 19

Table 4. Main harmonic components of the test TE.

Harmonic Amplitudes Harmonic Amplitudes Harmonic Amplitudes Components (Arcsec) Components (Arcsec) Components (Arcsec) 1 4.81 3n 2.44 6n 1.42 n 3.28 4n 1.56 8n 0.68 2n 2.68 5n 0.64 39n 0.47

Table 5. Percentage differences of analytical method relative to test results of backlash and TE caused by tooth profile modification (TPM) and load.

Parameters Analytical Method Test Results Diﬀerence (%) Backlash (arcsec) 28.61 31.03 7.79% TE (arcsec) 0.49 0.47 4.08%

2.3. Transmission Efficiency Because an RV reducer is a type of precision drive, the loss of the rotation angle is extremely small. The rotation angle loss can then be neglected, and only the torque loss owing to the friction torque is used to calculate the transmission efficiency. Because the friction loss of a bearing and load independent power loss is not the scope of the present study, it is not considered herein [28]. Therefore, the efficiency of the RV reducer is defined as follows:

T f ηc = 1 , (30) − To where T f is the friction torque of the cycloid gears, which can be calculated as follows:

Xm2 x y T = 2 F y0 + F x0 (31) f f i· pi f i· pi i=m1

2.4. Contact Ratio In theory, half of the cycloid teeth without a TPM will participate in the meshing simultaneously. However, because the cycloid profile will be thinned with a TPM, which inevitably reduces the meshing area, in order to study the variation in the meshing area caused by a TPM the contact ratio is defined, which may be calculated using the ratio of the actual meshing area length with the cycloid profile length of a single tooth. The contact ratio may be expressed as follows:

lc Rc = (32) l0

3. Effects of Modification Coefficients and Load

3.1. Tooth Profile Modification Method We applied three modification methods in this study. The first two methods are commonly used in engineering when forming a modification profile by changing the modification coefficients ∆rrp and ∆Rp. The cycloid profile of these two methods can be obtained using Equation (1). Case 1: Modification by fixing coefficient ∆rrp The modification coefficient ∆rrp is a constant in the case 1 method, which focuses on the impact of the modification coefficient ∆Rp on the RV reducer. Case 2: Modification by changing coefficients ∆rrp and ∆Rp with the same value In the case 2 method, the modification coefficients ∆rrp and ∆Rp are changed simultaneously to the same value. Appl. Sci. 2019, 9, 4099 14 of 19

Case 3: Modification based on traditional contact force formula The case 3 method is based on the characteristics of the contact force distribution along the cycloid profile, which uses the contact force curves as the modification curves. The case 3 method is defined as the reverse modification method in this study. The inverse modification curve is obtained using the traditional contact force formula of a cycloid gear [29]. The contact force of a cycloid profile may be calculated as follows: 1 2MS− 2 sin θi Pi = , (33) KZcRp Appl.where Sci.M 2019is, the9, x FOR load PEER torque REVIEW on the output shaft in the formula, but is a dimensionless coefficient14 usedof 19 to obtain different modification curves in this study. usingIn the a cycloidtraditional gear contact coordinate force system,formula the of a modification cycloid gear amount[29]. The is contact as follows: force of a cycloid profile may be calculated as follows: ( xsi = Pi cos(ψi ϕ) − (34) ysi𝑃= =Pi sin(ψi ϕ, ) (33) − whereThen, theM is reverse the load modification torque on cycloidthe output profile shaft is obtained in the formula, as: but is a dimensionless coefficient used to obtain different modification curves( in this study. x = x x In a cycloid gear coordinate system, the modificationf i i − si amount, is as follows: (35) y = y y f i i − si xPsi=− icos(ψϕ i ) where, x and y are the stand cycloid profile coordinates=−ψϕ in Equation (1). (34) i i yPsi isin( i ) Then, the initial tooth gap caused by a TPM using the case 3 method may be calculated as follows: Then, the reverse modification cycloid profile is obtained as: q 2 2 Ei(θi) = x + y (36)  x fiisi=−xxsi si  =− , (35) The modification parameters of the three methodsyyyfiisi are listed in Table6 based on the same maximum clearance of 0.012 to 0.105 mm. In addition, the TPM profiles with the same maximum modification where, 𝑥 and 𝑦 are the stand cycloid profile coordinates in Equation (1). gap are shown in Figure 11. To investigate the load effect, the transmission performances of an RV Then, the initial tooth gap caused by a TPM using the case 3 method may be calculated as follows: reducer under three load conditions were calculated respectively, including one to three times rated 22 output torque. Exyii()θ =+ sisi (36)

The modification parameters ofTable the 6. threeModiﬁcation methods conditions. are listed in Table 6 based on the same maximum clearance of 0.012 to 0.105 mm. In addition, the TPM profiles with the same maximum Case 1 Case 2 Case 3 modification gap are shown in Figure 11. To investigate the load effect, the transmission r R R = r M performances of an RV∆ reducerrp under three∆ p load conditions∆ p were∆ rp calculated respectively, including 0.01 mm 0.02 to 0.1 mm 0.01 to 0.225 mm 41 to 161 one to three times rated output torque.

Figure 11. Modiﬁcation curves. Figure 11. Modification curves. The transmission performance of an RV reducer includes three main aspects. The ﬁrst is the transmission accuracy, which canTable be measured 6. Modification based conditions. on the amplitude and mean value of the Case 1 Case 2 Case 3

𝛥𝑟 𝛥𝑅 Δ=ΔRp rrp 𝑀 0.01 mm 0.02 to 0.1 mm 0.01 to 0.225 mm 41 to 161 The transmission performance of an RV reducer includes three main aspects. The first is the transmission accuracy, which can be measured based on the amplitude and mean value of the transmission error. The second aspect is the mechanical performance, including the maximum normal contact force and maximum friction on the cycloid profile, which is obtained using the improved contact force formula proposed herein. The third aspect is the change in contact ratio and transmission efficiency.

3.2. Influence of Load for Case 1 Modification Method Appl. Sci. 2019, 9, 4099 15 of 19

transmission error. The second aspect is the mechanical performance, including the maximum normal contact force and maximum friction on the cycloid proﬁle, which is obtained using the improved contact force formula proposed herein. The third aspect is the change in contact ratio and transmission eﬃciency.

Appl.3.2. Sci. Inﬂuence 2019, 9, ofx FOR Load PEER forCase REVIEW 1 Modiﬁcation Method 15 of 19

ChangesChanges in thethe transmissiontransmission performance performance metrics metrics of anof an RV RV reducer reducer were were obtained obtained using using the case the 1 casemodification 1 modification method, method, as shown as shown in Figure in 12Figure. As can12. As be seencan be from seen Figure from 12 Figurea, an increase12a, an increase in the TPM in thesignificantly TPM significantly reduces thereduces contact the ratio. contact However, ratio. However, as the torque as the load torque increases, load increases, this effect this is eff effectectively is effectivelyreduced. Contrary reduced. to Contrary the changing to the trend changing of the trend contact of areathe contact shown area in Figure shown 12 a,in aFigure TPM will12a, increase a TPM willthe transmissionincrease the transmission efficiency of efficiency the RV reducer. of the RV The reducer. smaller The the loadsmaller torque, the load the moretorque, obvious the more the obviousincrease the in transmission increase in transmission efficiency, as efficiency, shown in Figure as shown 12b. in As Figure can be 12b. seen As from can Figure be seen 12 c,d,from the Figure TPM 12c,d,increases the theTPM maximum increases contact the maximum force and friction.contact Underforce and different friction. torque Under load conditions,different torque the change load conditions,trend is basically the change the same. trend At is the basically same time, the same. the TPM At the will same increase time, the the mean TPM value will ofincrease the TE, the as shownmean valuein Figure of the 12e. TE, However, as shown when in Figure the amount 12e. However, of modification when isthe less amount than 0.04 of mm,modification the TPM is reduces less than the 0.04amplitude mm, the of theTPM TE. reduces When thethe amount amplitude of modification of the TE. When is greater the thanamount 0.04 of mm, modification the TPM will is increasegreater thanthe amplitude 0.04 mm, the of the TPM TE, will as shownincrease in the Figure ampl 12itudef. of the TE, as shown in Figure 12f.

(a) (b) (c)

(d) (e) (f)

FigureFigure 12. 12. EffectEffect of of case case 1 modification 1 modification method method on onthe the transmission transmission performance performance of RV of RVreducer: reducer: (a) contact(a) contact ratio, ratio, (b) (transmissionb) transmission efficiency, efficiency, (c) (cmax) max contact contact force, force, (d ()d max) max friction, friction, (e ()e )mean mean TE TE value, value, andand ( (ff)) TE TE amplitude. amplitude.

3.3.3.3. Influence Influence of of Load Load for for Case Case 2 2 Modification Modification Method Method ChangesChanges in in the transmission performance metrics ofof anan RVRV reducerreducer areare obtained obtained using using the the case case 2 2modification modification method, method, as as shown shown in in FigureFigure 13 13.. ItIt cancan bebe seen from Figure 1313aa that the meshing area area underunder the ratedrated outputoutput torquetorque is is reduced reduced as as the the modification modification amount amount increases. increases. However, However, as the as loadthe loadtorque torque is increased is increased to twice to and twice three-times and three-ti the ratedmes torque,the rated the torque, meshing the area meshing increases area slightly increases rather slightlythan declining. rather than It can declining. be concluded It can that be theconc TPMluded does that not the always TPM reduce does not the meshingalways reduce area of the the meshingtooth profile area butof the is also tooth aff ectedprofile by but the is load also conditions.affected by However,the load conditions. as can be seen However, from Figure as can 13 beb, seenthe increasefrom Figure in the 13b, TPM the causes increase the in transmission the TPM caus effiesciency the transmission to decrease. efficiency This means to decrease. the TPM This will meansincrease the the TPM friction will torque,increase and the the friction smaller torque, the load and torque the smaller is, the the faster load the torque transmission is, the faster efficiency the transmissionwill decrease. efficiency In Figure 13willc, thedecr torqueease. loadIn Figure significantly 13c, the increases torque theload maximum significantly value increases of the contact the maximum value of the contact force and friction. However, the TPM causes the contact force and friction to decrease first, and then increase at a certain value of modification. Moreover, as can be seen from Figures 13c,d, the turning point is related to the load torque. Although the TPM increases the mean value of the TE, as shown in Figure 13e, the amplitude of the TE does not always increase with the TPM, particularly in the case of large loads. The results obtained in [9] also show that, if variables for compound modification are appropriately selected, a larger value of modification can still acquire smaller kinematic errors. It can be seen from Figure 13f that when the modification amount is less than 0.125 mm, the TPM has little effect on the amplitude Appl. Sci. 2019, 9, 4099 16 of 19 Appl. Sci. 2019, 9, x FOR PEER REVIEW 16 of 19 force and friction. However, the TPM causes the contact force and friction to decrease first, and then of the TE. When the modification amount is larger than 0.125 mm, the TPM causes the amplitude of increase at a certain value of modification. Moreover, as can be seen from Figure 13c,d, the turning the TE to significantly decrease at three-times the rated load. In addition, the amplitude of the TE at point is related to the load torque. the rated torque and at twice the rated torque is slightly increased.

(a) (b) (c)

(d) (e) (f)

Figure 13. 13. EffectEffect of of case case 2 2modification modification method method on onthe the transmission transmission performance performance of RV of RVreducer: reducer: (a) contact(a) contact ratio, ratio, (b) (transmissionb) transmission efficiency, efficiency, (c) ( cmax) max contact contact force, force, (d ()d )max max friction, friction, (e (e) )mean mean TE TE value, value, and ( f) TE amplitude.

3.4. InfluenceAlthough of Load the TPM under increases Case 3 Modification the mean valueMethod of the TE, as shown in Figure 13e, the amplitude of the TE does not always increase with the TPM, particularly in the case of large loads. The results Compared with the previous two traditional compound modification methods, the load effect obtained in [9] also show that, if variables for compound modification are appropriately selected, on the transmission performance using the case 3 method is specific. As can be seen from Figure a larger value of modification can still acquire smaller kinematic errors. It can be seen from Figure 13f 14a, the changing trend is basically the same as under both the rated load and thrice the rated load. that when the modification amount is less than 0.125 mm, the TPM has little effect on the amplitude of However, the TPM will cause an opposite changing trend at twice the rated load. Under the rated the TE. When the modification amount is larger than 0.125 mm, the TPM causes the amplitude of the load torque, the contact area is the largest when the modification coefficient is 101. With the TE to significantly decrease at three-times the rated load. In addition, the amplitude of the TE at the increase in the TPM, the meshing area is significantly increased at twice the rated load. For large rated torque and at twice the rated torque is slightly increased. load conditions, such as twice and thrice the rated load, the TPM has little effect on the transmission efficiency,3.4. Influence as ofshown Load underin Figure Case 14b. 3 Modification However, Methodthe TPM slightly reduces the transmission efficiency at the rated load. It can be seen from Figure 14c,d that, similar to the case 1 and case 2 methods, the TPM Comparedincreases withthe maximum the previous contact two traditional force and compound friction. As modification can be seen methods, from theFigure load 14e, effect the on influencethe transmission of the TPM performance on the mean using value the of case the 3 TE method is linear, is specific. as is the Asinfluence can be of seen the from load. Figure However, 14a, whenthe changing the modification trend is basically coefficient the same is 86 asat underthe rate bothd torque, the rated the load amplitude and thrice of the ratedTE is load.minimized, However, as shownthe TPM in willFigure cause 14f. an The opposite TPM causes changing the trendTE amplit at twiceude theto increase rated load. slightly Under at thetwice rated the loadrated torque, load. Inthe addition, contact areathe TE is theamplitude largest whenis slightly the modification reduced at three coeffi timescient the is 101. rated With load. the increase in the TPM, the meshing area is significantly increased at twice the rated load. For large load conditions, such as twice and thrice the rated load, the TPM has little effect on the transmission efficiency, as shown in Figure 14b. However, the TPM slightly reduces the transmission efficiency at the rated load. It can be seen from Figure 14c,d that, similar to the case 1 and case 2 methods, the TPM increases the maximum contact force and friction. As can be seen from Figure 14e, the influence of the TPM on the mean value of the TE is linear, as is the influence of the load. However, when the modification coefficient is 86 at the rated torque, the amplitude of the TE is minimized, as shown in Figure 14f. The TPM causes the TE amplitude to increase slightly at twice the rated load. In addition, the TE amplitude is slightly reduced (a) (b) (c) at three times the rated load. Appl. Sci. 2019, 9, x FOR PEER REVIEW 16 of 19 of the TE. When the modification amount is larger than 0.125 mm, the TPM causes the amplitude of the TE to significantly decrease at three-times the rated load. In addition, the amplitude of the TE at the rated torque and at twice the rated torque is slightly increased.

(a) (b) (c)

(d) (e) (f)

Figure 13. Effect of case 2 modification method on the transmission performance of RV reducer: (a) contact ratio, (b) transmission efficiency, (c) max contact force, (d) max friction, (e) mean TE value, and (f) TE amplitude.

3.4. Influence of Load under Case 3 Modification Method Compared with the previous two traditional compound modification methods, the load effect on the transmission performance using the case 3 method is specific. As can be seen from Figure 14a, the changing trend is basically the same as under both the rated load and thrice the rated load. However, the TPM will cause an opposite changing trend at twice the rated load. Under the rated load torque, the contact area is the largest when the modification coefficient is 101. With the increase in the TPM, the meshing area is significantly increased at twice the rated load. For large load conditions, such as twice and thrice the rated load, the TPM has little effect on the transmission efficiency, as shown in Figure 14b. However, the TPM slightly reduces the transmission efficiency at the rated load. It can be seen from Figure 14c,d that, similar to the case 1 and case 2 methods, the TPM increases the maximum contact force and friction. As can be seen from Figure 14e, the influence of the TPM on the mean value of the TE is linear, as is the influence of the load. However, when the modification coefficient is 86 at the rated torque, the amplitude of the TE is minimized, Appl. Sci. 2019, 9, 4099 17 of 19 as shown in Figure 14f. The TPM causes the TE amplitude to increase slightly at tw

Appl. Sci. 2019, 9, x FOR PEER REVIEW 17 of 19 (a) (b) (c)

(d) (e) (f)

Figure 14. EffectEffect of of case case 3 3modification modification method method on on the the transmission transmission performance performance of RV of RVreducer: reducer: (a) (contacta) contact ratio, ratio, (b) (btransmission) transmission efficiency, efficiency, (c) ( cmax) max contact contact force, force, (d (d) )max max friction, friction, ( (ee)) mean mean TE TE value, value, and (f) TE amplitude.

4. Discussion To determine the optimal profileprofile of the modificationmodification methods used for different different applications, the changeschanges inin the the mean mean value value and and amplitude amplitude of theof the TE, TE, the contactthe contact ratio, ratio, the maximum the maximum contact contact force, theforce, friction, the friction, and the and transmission the transmission efficiency efficiency of the three of the kinds three of modificationkinds of modification methods are methods compared. are Thecompared. TPM increases The TPM the increases mean value the of mean the TE value linearly of underthe TE the linearly three load under conditions the three using load the conditions proposed threeusing modificationthe proposed methods. three modification However, themethods. mean value However, of the the TE mean is at minimum value of the when TE usingis at minimum the case 2 methodwhen using as compared the case 2 with method that ofas thecompared other two with methods. that of the In addition,other two the methods. mean value In addition, of the TE the is themean largest value when of the using TE theis the case largest 1 method. when However, using the under case 1 large method. load conditions,However, under the amplitude large load of theconditions, TE is the the smallest amplitude when of using the TE the is case the 3smallest method. when Moreover, using the case 13 methodmethod. willMoreover, reduce the contactcase 1 method ratio significantly will reduce from the 0.38 contact to 0.24 ratio under sign theificantly rated loadfrom condition, 0.38 to 0.24 although under the the contact rated ratioload changescondition, slightly although when the using contact the caseratio 2 andchanges 3 methods. slightly The when maximum using the contact case force 2 and and 3 frictionmethods. have The a changingmaximum point contact when force using and the friction case 2 method. have a Moreover,changing thepoint TPM when has littleusing eff ectthe oncase the 2 maximum method. contactMoreover, force the and TPM friction has little when effect using on the the case maximu 3 method,m contact although force the and eff frictionect is significant when using when the using case the3 method, case 1 method. although Although the effect the is firstsignificant two modification when using methods the case will 1 method. affect the Although transmission the efirstfficiency two significantly,modification methods the case 3will method affect has the atransmission slight effect onefficiency the transmission significantly, effi ciency,the case with 3 method a change has ofa onlyslight 92.8% effect toon 92.1% the transmission under the rated efficiency, load condition. with a change Therefore, of only it can92.8% be concludedto 92.1% under that thethe caserated 2 andload 3condition. modification Therefore, curves areit can better be concluded than that of that the the case case 1 method 2 and 3 in modification terms of the contactcurves are force better and transmissionthan that of the accuracy. case 1 However,method in under terms large of the load contact conditions, force and such transmission as twice the ratedaccuracy. load, However, the case 3 methodunder large may load be the conditions, optimal profile such as modification twice the rated method load, compared the case with3 method the case may 2 method.be the optimal This is becauseprofile modification the case 3 method method can compared maintain awith good the transmission case 2 method. performance This is because with an increasethe case in3 themethod load andcan modificationmaintain a good coeffi cient.transmission performance with an increase in the load and modification coefficient. 5. Conclusions 5. ConclusionsOwing to the multi-tooth contact that occurs in an RV reducer, the results calculated using a traditionalOwing contact to the modelmulti-tooth with a singlecontact contact that occurs stiffness in willan RV be inaccurate.reducer, the Therefore, results calculated a new multi-tooth using a contacttraditional model contact and a transmissionmodel with errora single model contact of an RVstiffness reducer will were be proposed inaccurate. to analyze Therefore, the influence a new ofmulti-tooth the tooth profilecontact modificationmodel and anda transmission load on the error transmission model of accuracy, an RV mechanicalreducer were properties, proposed and to transmissionanalyze the influence efficiency. of The the following tooth profile conclusions modification were drawn: and load on the transmission accuracy, mechanical properties, and transmission efficiency. The following conclusions were drawn: (1) The accuracies of the contact force model and the transmission error model were validated by comparing the calculation results of the FEM and the test results of the RV reducer prototype, respectively. (2) In general, a tooth profile modification and load have a comprehensive effect on an RV reducer. Both the modification and load increase the mean value of the transmission error, although the appropriate modification amount can be selected to minimize the amplitude of the transmission error under different loads. With modification method 2, the contact force and friction can be minimized under the optimal modification amount, which increases linearly Appl. Sci. 2019, 9, 4099 18 of 19

(1) The accuracies of the contact force model and the transmission error model were validated by comparing the calculation results of the FEM and the test results of the RV reducer prototype, respectively. (2) In general, a tooth profile modification and load have a comprehensive effect on an RV reducer. Both the modification and load increase the mean value of the transmission error, although the appropriate modification amount can be selected to minimize the amplitude of the transmission error under different loads. With modification method 2, the contact force and friction can be minimized under the optimal modification amount, which increases linearly with the load. When the load is large, increasing the modification amount will increase the contact ratio for modification methods 2 and 3. (3) Modification method 1 will significantly reduce the contact ratio and increase the contact force, although the friction torque will be reduced, thus improving the transmission efficiency. In general, modification method 2 achieves a good performance and is suitable for common modification occasions. The transmission efficiency and contact ratio of modification method 3 are almost unaffected by the modification, and thus the method is suitable when a large clearance and load are needed.

The multi-tooth contact model presented here can also be applied to various proﬁle modiﬁcation methods. Moreover, this research can serve as a fundamental for both the analyses of the transmission characteristics of the RV reducers, and further investigations of the rigidity and strength of the RV reducers.

Author Contributions: Conceptualization, Z.-Y.S. and H.W.; methodology, Z.-Y.S. and H.W.; validation, B.Y. and H.W.; formal analysis, H.W.; data curation, H.X.; writing—original draft preparation, H.W.; writing—review and editing, H.W. and B.Y.; funding acquisition, Z.-Y.S. and B.Y. Acknowledgments: This research was supported by the National Natural Science Foundation of China (No. 51805011, 51635001), and the National High Technology Research and Development Program 863 of China (No. 2015AA043002). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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